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Properties

Label 20.2.52780750333...4375.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,5^{10}\cdot 13^{4}\cdot 41^{4}\cdot 669679$
Root discriminant $15.35$
Ramified primes $5, 13, 41, 669679$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, 5, 7, 9, -55, -17, 80, 23, -24, -17, -14, 33, 12, -16, 6, -6, 2, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 5*x^18 + 2*x^17 - 6*x^16 + 6*x^15 - 16*x^14 + 12*x^13 + 33*x^12 - 14*x^11 - 17*x^10 - 24*x^9 + 23*x^8 + 80*x^7 - 17*x^6 - 55*x^5 + 9*x^4 + 7*x^3 + 5*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 5*x^18 + 2*x^17 - 6*x^16 + 6*x^15 - 16*x^14 + 12*x^13 + 33*x^12 - 14*x^11 - 17*x^10 - 24*x^9 + 23*x^8 + 80*x^7 - 17*x^6 - 55*x^5 + 9*x^4 + 7*x^3 + 5*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 5 x^{18} + 2 x^{17} - 6 x^{16} + 6 x^{15} - 16 x^{14} + 12 x^{13} + 33 x^{12} - 14 x^{11} - 17 x^{10} - 24 x^{9} + 23 x^{8} + 80 x^{7} - 17 x^{6} - 55 x^{5} + 9 x^{4} + 7 x^{3} + 5 x^{2} + 10 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-527807503333353115234375=-\,5^{10}\cdot 13^{4}\cdot 41^{4}\cdot 669679\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.35$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 13, 41, 669679$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{79} a^{17} + \frac{33}{79} a^{16} - \frac{9}{79} a^{15} + \frac{30}{79} a^{14} - \frac{23}{79} a^{13} + \frac{17}{79} a^{12} - \frac{1}{79} a^{11} + \frac{35}{79} a^{10} + \frac{15}{79} a^{9} + \frac{19}{79} a^{8} + \frac{11}{79} a^{7} - \frac{27}{79} a^{6} + \frac{27}{79} a^{5} - \frac{19}{79} a^{4} + \frac{39}{79} a^{3} - \frac{8}{79} a^{2} + \frac{2}{79} a - \frac{11}{79}$, $\frac{1}{79} a^{18} + \frac{8}{79} a^{16} + \frac{11}{79} a^{15} + \frac{14}{79} a^{14} - \frac{14}{79} a^{13} - \frac{9}{79} a^{12} - \frac{11}{79} a^{11} - \frac{34}{79} a^{10} - \frac{2}{79} a^{9} + \frac{16}{79} a^{8} + \frac{5}{79} a^{7} - \frac{30}{79} a^{6} + \frac{38}{79} a^{5} + \frac{34}{79} a^{4} - \frac{31}{79} a^{3} + \frac{29}{79} a^{2} + \frac{2}{79} a - \frac{32}{79}$, $\frac{1}{668409413429} a^{19} - \frac{1223458740}{668409413429} a^{18} - \frac{3347679057}{668409413429} a^{17} - \frac{119362043807}{668409413429} a^{16} - \frac{315550296525}{668409413429} a^{15} - \frac{111463163488}{668409413429} a^{14} - \frac{141882061163}{668409413429} a^{13} - \frac{13238977758}{668409413429} a^{12} + \frac{182296600919}{668409413429} a^{11} + \frac{46943243955}{668409413429} a^{10} - \frac{243530263186}{668409413429} a^{9} + \frac{32463236412}{668409413429} a^{8} + \frac{101900398112}{668409413429} a^{7} - \frac{163642600422}{668409413429} a^{6} - \frac{308207067060}{668409413429} a^{5} + \frac{15175765368}{668409413429} a^{4} - \frac{295935536522}{668409413429} a^{3} - \frac{12540888129}{668409413429} a^{2} - \frac{158936951939}{668409413429} a + \frac{46971208764}{668409413429}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4535.84075701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
669679Data not computed